By Professor Dr. Berndt Müller, Dr. Joachim Reinhardt (auth.)
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Extra resources for Neural Networks: An Introduction
15) An explicit expression is obtained by means of the Fourier transform p(k, 1) == J dxp(x, l)exp(ikx) = ~[exp(ik) + exp(-ik») = cos(k) . e. p(k, N) p(k, N - l)P(k, 1), we find that p(x, N) = 2111" J = dk p(k, N) exp( -ikx) = 2111" J dk cos N (k) exp( -ikx) . 17) An approximate expression for large N can be found by noting that this limit corresponds to small values of k, where we can expand the cosine function: cosN (k) R: (1 - Finally p(x, N) R: 2~ ~k2)N R: exp ( -~Nk2) J dk exp(-ikx - . 18) ~Nk2) = (211"N)-1/2 exp(;~) showing that the distribution has the width L1x oc ,fFi.
2, replacing the quantities Vj, nj by new variables aj, Sj, which are defined as vf. vt Si = 2nj - 1, aj = 2vj - 1. 2) The new variables take the values ±1, instead of 0 and 1. The squared deviations are then expressed as and the search for the minimum of H/1 is then equivalent to the search for the maximal value of the function N A/1(sj) = I~>fsj (f-l = 1, ... 4) i=l for a given pattern Sj. L is called the (Euclidean) scalar product or overlap of the N-component vectors and Sj. According to Hopfield [H082] the dynamical evolution of the state of the network is now defined as follows.
Temperature and threshold zero, and experiment with the updating prescription: = random; 1 = sequential; 2 = synchronous. 4 Neural "Motion Pictures" Temporal sequences of recalled memories form a very important aspect of our mental capacity. 4 Neural "Motion Pictures" 33 actions, which otherwise would consist of meaningless single pictures. Periodic sequences of neural impulses are also of fundamental importance for the control of motoric body functions, such as the heartbeat, which occurs with great regularity almost three billion times during an average person's life.